I'm having trouble figuring out the response of this circuit, specifically the voltage across the capacitor in my homework. The input from the source is a unit step function, and there are no initial conditions for the capacitor or inductor.
simulate this circuit – Schematic created using CircuitLab
I first turned it into a norton equivalent, and used node analysis to arrive at the differential equation.
$$\frac{d^2}{dt}e_1+\frac{d}{dt}e_1(\frac{R_2}{L}+\frac{1}{R_1C})+e_1(\frac{1}{LC}+\frac{R_2}{CLR_1})=\frac{d}{dt}\frac{R_2}{C}I_1+\frac{R_2}{LC}I_1$$
I know that the solution to this equation takes the form:
$$ e_1(t) = Ae^{-\alpha t}cos(\omega_dt+\phi)$$
So I think my next step is to find A and phi by the following expressions:
$$e_1(0)= ?$$
$$\frac{d}{dt}e_1(0) = ?$$
However, I am a bit unsure about how exactly to go about this. My reasoning suggests that they are both 0 - at the moment the source switches from 0v to 1v, the capacitor is a short circuit and the inductor is an open circuit. So all the current flows through the capacitor, and there is no voltage drop since the current source is shorted - so \$e_1 = 0\$ and \$e_1'=0\$. However, this means that A and phi are also both zero, so this is confusing for me. Can someone give me some help with finding A and phi?